Exact Dynamics

72

IEEE TRANSACTIONS ON COMMUNICATIONS, JANUARY 1974

sampling interval which in every case is almost identical with the results of the simulation. This holds true even for the low SNR. Thusincriticalapplications,thesecurves,whenmodified t o reflect the correct message bandwidth, could be used to determinetheminimumsamplingrate. In reality, of course, theactualvalue o f theerrorvarianceatthreshold is in the vicinityof 0.25. However,thecomputationaldifficultyin applying more a accurateapproximation of the variance equation is notjustifiablesince all thatwouldbegained essentially is anincreaseintheverticalslopeofthecurvesin Figs. 3-5. Theconsistencyoftheaforementionedresults precludes the need for this added difficulty.

CONCLUSIONS The conditions have been established which insure the validity of the,white uniform sequence model for quantization SNR is defined which allows error in the! DPLL. An effective quantized system performance to be predicted from unquantizedresults.Samplingrequirementshavebeenexperimentally determined through simulation. A method has been described to predict these minimum sampling rates using the high SNR variancerelations.Samplingrequirementsforthe quantized system may then be determined using the effective

REFERENCES S. C. Gupta,“Onoptimumdigitalphase-lockedloops,” IEEE Trans. Commun. Technol. (ConcisePapers),vol.COM-16,pp. 340-344, Apr. 1968. G . PasternackandR. L. Whalin,“Analysisandsynthesisofa digital phase-locked loop for FM demodulation,” Bell Sysf. Tech. J . , vol. 47, pp. 2207-2237, Dec. 1968. S. C. Gupta, “Status of digital phase-locked loops,” in Proc. 3rd Hawaii Int. Conf., pp. 255-259, 1970. J. Garodnick, J . Greco, and D. L. Schilling, “An all digital phaselockedloopfor FM demodulation,” in Proc. Int. CommunicationsConf., June 1971. J . K. Holmes, “Performance of a first-order transition sampling digital phase-locked loop using random-walk models,” IEEE Trans. Commun., vol. COM-20, pp. 119-131,Apr. 1972. J . R . Cessna and D. M. Levy, “Phase noise and transient times for abinaryquantizeddigitalphase-lockedloopinwhiteGaussian noise,” IEEE Trans. Commun., vol.COM-20,pp.‘94-104,Apr. 1972. G. S. Gill and S. C. Gupta,“First-orderdiscretephase-locked loop with applications to demodulation of’angle-modulated carrier,” IEEE Trans. Commun. (Concise Papers), vol. COM-20, pp. 454-462, June 1972. C. N. Kelly and S. C. Gupta, “The digital phase-locked loop as a near-optimum FM demodulator,” IEEE Trans. Commun. (Concise Papers), vol. COM-20, pp. 406-411, June 1972. - “Discrete-time demodulation of continuous-timesignals,” IEkE Trans. Inform. Theory, vol. IT-18, pp. 488-493, July 1972. G. S. Gill and S. C. Gupta, “On higher order discrete phase-locked loops,” IEEE Trans. Aerosp.Electron. Syst., vol. AES-8,pp. 615-623, Sept. 1972. C. P. Reddy and S. C. Gupta, “Demodulation of FM signals by a in Proc. Int. Telecommunications discretephase-lockedloop,” Con$, Los Angeles, Calif.,Oct. 1972. -, “A class of a l l digitalphaselockedloops:Modelingand analysis,” IEEE Trans. Ind. Electron. Confr. Instrum., vol. IECI-20, pp. 239-25 1 , Nov. 1973. D. R.Polkand S. C. Guta,“Quasi-optimumdigitalphase75locked loops,” IEEE Trans. Commun., vol.COM-21,pp. 82, J y . 1973. -, Anapproachto.theanalysisofperformanceofquasiIEEE Trans. Commun., optimum digitalphase-lockedloops,” vol. COM-21, pp. 733-738, June 1973. S. C. Gupta, “On theperformanceof digital G.T.Hurstand phase-locked loops in the threshold region,” to be published. The Stare’ Variable Approach t o Continuous D. G.Snyder, Estimation, Res. Mono. 5 1. Cambridge, Mass.: M.I.T. Press, 1969. B. Widrow, “A study of rough amplitude quantization by means ofNyquistsamplingtheory,” IRETrans.CircuitTheory, vol. CT-3, pp. 266-276, Dec. 1956.. W. R. Bennett, “SPECTRA of quantized signals,” Bell Sysr. Tech. J . , VOI. 27, pp. 446-471, July 1948. A. Papoulis, Probability, Random Variables and Stochastic Processes. New York: McGraw-Hill, 1964, p. 133. G. T. Hurst, “Sampling, quantizing, and low signal to noise ratio considerations in digital phase-locked loops,” Ph.D. dissertation, Southern Methodist Univ., Dallas, Tex., Apr. 1973.

P

Exact Dynamics of Automatic Gain Control JOHN

E.

OHLSON, MEMBER, I E E ~

Abstract-The exact input-output ielationship is derived for a fustorder automatic gain control loop wherein the variable gain is an exponential function of the gain control voltage. T h e exact solution is compared to the linearized solution, and the condition for valid

linearization is given.

I. INTRODUCTION Automatic gain control (AGC) loops are used in virtually all modern communication systems. The work of Oliver [ 11 and Victor and Brockman [ 2 ] provided a useful theory both for static and small signal analyses. In some applicatiqns, howeverj a more general theory is required which can predict what will happen when large variations in signal level occur. Examples of large variations in signal level include severe fading in urban mobile links and links to/from tumbling satellites. When large signal level variations occur,.the linear AGC, theory is n o longer useful, and we must attempt to solve the nonlinear problem. Fig. 1 illustrates the AGC problem where the output y ( t ) is given by

L ( t )= g ( u ) x ( t )

(1)

where x([) is the input, ~ ( tis)the gain control voltage, g ( u ) is the gain control characteristic which is a memoryless function of u ( t ) , and b 0 is the AGC reference bias. Seseralworkers [ 3 ] - [ 6 ] assumedthegaincontrolcharacof w , and have obtained some teristic to be a linear function interesting exact results. Plotkin [ 7 ] assumed g ( u ) t o be of the form Y O 1 , where a is a constant, and obtained a perturbation solution. However, neither a linear nor a u-“ characteristic is representative of typicalgain-controlledamplifiers.Usually these amplifiers must have a gain dynamic range of 50-100 dB. For practical implementation, it has been found that a gain whichvariesexponentiallywith u gives the desired dynamic range with a moderate range of u, and is also easy to characterize $rice an exponential g ( u ) gives gain in decibels as a linear function of gain control voltage u. Victor and Brockman [ 2 ] assumedthisformfor g ( u ) , andsubsequentlyalmostall modern receiver designs have approximated this characteristic. The analysis to follow thus applies to alarge number of current systems.

>

11. ANALYSIS We shall assume on the basis of the above that g ( u ) varies as

>

>

where G 0 and a 0 are constants. Since gain, the gain in decibels is a linear function above :

g(u) is a voltage

of u as discussed

g ( u ) in decibels = 20 loglo g ( u ) = G (in decibels)

-

( 8 . 6 8 6 ~ ~ ) ~ . (3)

We mustnowspecifytheAGCloopfilter.Inmostactual systems, the loop filter is asimplelow-pass R C filter.However, the R C time constant is usually so much longer than the closed-loop response time that the loop filter can be approximated excellently by a simple integrator. This we do, and the systemweshallanalyzeisshowninFig. 2. We includethe Paper approved by the Associate Editor for Communication Theory of for publicationwithoutoralpretheIEEECommunicationsSociety 2 8 , 1973. sentation. Manuscript received January The author is with the Department of Electrical Engineering, Naval Postgraduate School, Monterey, Calif. 93940.

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CONCISE PAPERS

x(t

1

INPUT

13

QAlN 0 (VI

-

g(v) x ( t i

v

'

--

GAIN I(l)

Y(t)

v

-aV

Ge

OUTPUT

,Y(l)

Ah

A

V(l)

+ dl

v(t)

GAIN-CONTROL VOLTAGE

A GC LOW FILTER

0

A

4----

-

K &b

;(I)

7

BIAS "0

Fig. 1. AGC block diagram.

Fig. 2. First-order AGC loop.

quantity uo as the initial value of u ( t ) at t = 0, i.e., ~ ( 0=) U O . Note also that we incorporate an open-loop gain factor K > 0. We nowwish to find y ( t ) and u ( t ) in terms of x ( t ) . From Fig. 2 it is clear that

d = K [ y - b ] = K (xCe-"'

-

g = -&gd,

g

1 u(t)=C+-log, a

b) (4)

of t in our notation, and a dot where we drop explicit use denotes time derivative. From (2) we can see that

andbysubstitutingfor obtain

conditiondisappears. We may thus drop the transient exponential terms in ( 10) and ( 1 1) and consider the "stationary" case where the gain control voltageis

d in(4)andbyuse

[h*x(t)l

(12)

where the constant C i s 1 C = - log, ( G / b )

(5)

(13)

a!

of (2),wecan and the output is

+ Kaxg'

- K a ! b g = 0.

(6) b x(t) Y ( t ) = ___ h * x(t)

This is intheform of Bernoulli'sequationforwhichthe solution is known to be [ 81

(14) '

The results in (12) and (14) are the principal contribution of this work. It must be noted that h * x ( t ) must remain positive or (12) and (14) make little sense. Certainly if x ( t ) 0 , then h * x ( t ) 0 and there is no difficulty.However; if x ( t ) is where r = l/Ka!b. Since attenuation is the inverse of gain, we negative often enough so that h * x ( t ) + 0 , then the system see that the voltage-controlled attenuation is a Iinear function will achieve the state u = --M from which it cannot recover. It of the input plus a decaying initial condition. The integral in is easy to appreciate this difficultyif the loop is assumed to be (7) is recognized as a convolution 12 * x ( t )where in steady state due to x ( t ) = Ixl 1, say, and then let x ( t ) switch abruptly to x ( t ) = - Ix2 1. It can then be shown that ~ ( t-+ ) --oo in a finite time, and once there, recovery is not possible. In noncoherent receivers, x ( t ) is an envelope function, and hence alwayshasthesamesign, so thereis no problemthere. In t 0), Fig. 3 is an equivalent model of the AGC loop. It is interesting that the highly nonlinear AGC loop of Fig. 2 (10) hasthesimpleequivalentmodel of Fig. 3. It is indeedreof the input x ( t ) by h ( t ) is markablethatalinearfiltering Also, since y = x g , we use (9) and have centraltotheloopoperation.Italso is intuitivelypleasing that y ( t ) is proportional to x ( t ) normalizedby an averaged x(t).

>

>

>

,

111. THE STATIONARY CASE The case of greatest practical interest is when the system has operatedforalongtimeanddependenceupontheinitial

Iv. COMPARISON W I T H THE LINEARIZED MODEL Since we have the exact dynamics of the loop, we shall now consider how good a linearized analysis can be. Consider the static case with x ( t ) = x o> 0. We clearly obtain h * x ( t ) = x0 > so

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74

IEEE TRANSACTIONS ON COMMUNICATIONS, JANUARY 1974

.

Ap<< 1 rather than would require only the weak condition A 1. It is also easyt o see the effects of “fast” AGC (UT 1)‘and“slow”AGC(UT>> 1 ) whereweget y ( t ) b and y ( t ) 2 b ( 1 + A sin ut), respectively. Another interesting property of the loop is seen in (23). We see there that although u ( t ) may be a sdverely distorted version of x ( t ) , its times of maxima and minima do n.ot depend upon the amplitude A A of the sinusoidal component of x ( t ) . That the “phase shift” is not dependent ‘upon signal amplitude is a somewhat surprising property considering the nonlinearity of of theloop.Forsmallvariations,thepropertyisexpected, course, but for large variations, it would not seem obvious and to is an interesting property not previously known. It is easy see that this “phase-shift’’ invariance generalizes to any periodic variation forx ( t ) .

<<

f

loac (G/b)

Fig. 3. Equivalent AGC loop model.

u(t) =

1 c +log, a

x0

VI.NOISE ANALYSIS

and y ( t )= b. Thus the static casegives the gain control voltage as alogarithmicfunction of thesignal,andtheoutputis stabilized exactly to the value of the AGC reference bias b . If the signal now varies aboutx, i.e.,

~ ( t= x) 0

<<

+A(t),

the exact u ( t ) and y ( t ) are

Althoughwe have the exact solution for y ( t ) and u ( t ) in terms of x ( t ) , i t is difficult to obtain exact results for excitation of the loop by random processes. The difficuity, of course, is the requirement that h * x ( t ) 0. An approximate solution can be obtained if the input is modeled as apositivesignal plus a weak noise component. Then the output noise contribution clearly can, be obtained by a linear perturbation of the I\ solution due to signal alone. In one case we can obtain thkp$$ first-order probability density of y ( t ) when x ( t ) is an arl$t&ry Gaussian random process. First note that h * x . ( t ) has d-finite probability of being negative,regardless of how large apositivemean x ( t ) has. Thus, in theory, our loop can get stuck at u = --oo.However,’in practice,finitedynamicrangeswill’.preventthis, so we may consider that we are evaluating y ( t ) from the model in Fig. 3, which cannot get stuck. We ignore the apparent mathematical difficulty of u ( t ) beinginfinite‘orimaginary,using-the dynamic range argument above. If x ( t ) is Gaussian, then so is h * x ( t ) , andthus y ( t ) isaratiooftwoGaussianrandom variableswhosemeans,variances,andcovarianceareeasily calculated.Theprobabilitydensityfunction is verycomplicated and will not be given here. The interested reader may consult Omura and Kailath’s widely distributed compendium [9] for the form of the density function. There is a great interest in being able to statistically characterize y ( t ) and u ( t ) when x ( t ) is Rayleigh,Rician,or lognormally distributed, corresponding to noncoherent receivers. The difficulty is that to statistically analyze the loop, we need morethanjustfirst-orderstatistics o f x ( t ) , which isall the above distributions provide.

>

’

1-

.

x.

J

\ -

A(t)

1 +-

(18) X0

If h

* A ( t ) << x o , then

obvious approximations give log xo

u(t) = c + e ct!

+

h

* A(t)

(19)

X0

Theseexpressionsfor u ( t ) and y ( t ) are exactly what is obtained when the gain function e-au(t) is linearized around the is that we do not static value of u in (1 5). What we have shown require A c t ) <<xo for linearization to be valid, but only the much weaker conditionh * A ( t ) << x o . For an example, consider the extreme case where x ( t ) has a very large sinusoidalvariablecomponentcomparedtoits “nominal” value, e.g., x(t) = 1

+ 100 sin ut.

(21)

Without knowing the exact solution of (17) and (18), linearization would appear out of the question under any conditionbecause we certainly d o n o t have a small-signal variation. How~ h * ( 1 0 0 sin ut) l, ever, if w is larger enough than l / so we have the remarkable result that the linearized results still apply.

<<

’?

VII. CONCLUSION We have modeled a very common AGC loop as a first-order nonlinear system, and have found the exact response for essentially arbitrary input signal. The solution has the interesting property that the primary operation upon the signal is a linear filteringoperation.This is thenfollowedbyamemoryless nonlinearity. A further property of interest is that linearization of the loop operation isvalid under a much less restrictive condition t,han previously assumed.

V. EXAMPLE

Let the input be x ( t ) = A ( 1 + A sin at) where A and A are constants. It is easy to show that 1 + A sin ut v ( t )= b 1 + A[ sin (ut+ 0 )

1

1 A + - l o & [ 1 + A D sin (ut+ e ) ] (23) a where 0 = [ 1 + ( ~ 7] ) ~ and 0 = - tan-’(uT).Inthisexof how a valid linearization ample we observe a specific case ,U(t) = C

+ - log, ct!

REFERENCES B. M . Oliver, “Automatic volume control as a feedback problem,” Proc. IRE, vol. 3 6 , pp. 466-473, Apr. 1948. W. K. Victorand M. H. Brockman,“Theapplication o f linear servotheory to the’design of AGC loops,” Proc.IRE, vol. 48, pp. 234-238, Feb. 1960. E. D. Banta, “Analysis o f an automatic gain control (AGC),” IEEE Trans. Automat. Conrr.,vol. AC-9, pp. 181-182,Apr. 1964. H. SchachterandL.Bergstein,“Noiseanalysis of anautomatic gaincontrolsystem,” IEEE Trans. Automat.Contr., vol.AC-9, pp. 249-255, July 1964. W. J . Gill and W. K. S. Leong, “Response of’an AGC amplifier to two narrow-band input signals,” IEEE Trans. Commun. Technol;, vol. COM-14, pp. 4 0 7 - 4 1 7 , Aug. 1966.

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75

CONCISE PAPERS AGC inan AM-FM telemetry system,” IEEE Trans. Commun. Tecknol., vol. COM-18, pp. 59-63, Feb. 1970. 171 S. Plotkin, “On nonlinearAGC,” Proc. IEEE (Corresp.),vol. 5 1 , p. 380, Feb. 1963. 181 K. Rektorys, Survey of Applicable Mafhentatics. Cambridge, Mass.: M.I.T. Press, 1969, pp. 746, 834. 191 J . Omura and T. Kailath, “Some useful probability distributions,” StanfordElectron.Lab.,Stanford,Calif.,Tech.Rep.7050-6, SU-SEL-65-079, Sept. 1965, pp. 47-49. [ 6 ] R. S. Simpson and W. H. Tranter, “Baseband

0

T

companding

5

overload

0,

> Y

0,

E

-2ov 0

Digital Companding Techniques C. J. KIKKERT, MEMBER, IEEE

20

\ I

LO 60 80 Relative inputindB. Fig. 1 . Comparison between the performance of companded and uncompanded code modulators.

Syllabiccompandingcanbedividedintotwotypes [91: Abstuact-This paper deals with the requirements for t h e design of 1)compandingwithincompletecontroland 2) companding digital companding techniques in either delta or pulse-code modulation. with complete control. Companding with incomplete control Both delta and pulse-code modulation convert analogue signals into stretches a region of the SNR curve in Fig. ](a) horizontally binary signals and in both these systems the dynamic range is normally while companding with complete control stretches one point small. By t h e use of companding, the dynamic range can be extended. ontheSNR curvehorizontally t o give Fig.I(b).Itcanbe Since both’ delta and pulse-code modulation are digital methods, they seen that companding with complete control gives a flat SNR are well suited to theuse of digital companding techniques. gives a of t h e overtheentirecompandingrangeandconsequently Thebinarytransmittedsignalnormallycontainsameasure better performance than companding with incomplete control. systemperformance.Byobservingcertainpatternsinthisbinary It is possible to apply syllabic companding to a code modulasignal and using the occurrence or nonoccurrence of these patterns to as well, SO change thegainof the modulator and demodulator, syllabic companding tion system which has instantaneous companding can be obtained. The selection of the binary pattern and the rate of that double companding can be obtained. Therehavebeenmanysyllabiccompandingschemesfor change of gain of the modulator and demodulator, determines both the [IO]-[ 141. Due delta modulation, using analogue techniques point at which the companding operates and the attack and decay times. tohardwarelimitationsonlycompandingwithincomplete The ratio of the largest t o t h e smallest value of the gain determines control has been obtained. thedynamicrange. By t h e use of digitalcircuitry,thegaincanbe In order to obtain companding with complete control, either controlled with sufficient accuracy over a large dynamic range. a separatesignalcontainingtheinformationrelatedtothe The paper deals with the principles involved in selecting t h e binary inputpowermustbesent [ 1 5 1 , [ 161 ordigitaltechniques patterns to control the gain of the modulator and as examples a delta must be used [ 171-[20]. The last two papers use a one bit modulation system and a pulse-code modulation system with companding ratios of 60 dB are discussed.

I. INTRODUCTION Code modulation is a group of modulation methods where theanalogueinput is sampledandwhereateachsampling instant a code word representing the input is generated. PCM, APCM, AM and their variants are examples of code modulation systems. Since code modulation approximates the input signal, distortion called quantization distortion is introduced. A typical plot of the resulting signal-to-quantization-noise ratio (SNR) versus input signal is shown in Fig. l(a). It can be seen that a high SNR is only obtained for a narrow range of input signals. Companding is used t o increasethedynamicrange over which a high SNR occurs. There are two basic types of 2 ) syllabic companding:1)instantaneouscompandingand companding. Instantaneous companding has been applied by many workers t o b o t h PCM [ 1 ] -[ 3 I and delta modulation [41-[ 8 I . Instantaneous companding alters the shape of the SNR curve andthepeakSNRmayevenbehigherthanfortheuncompanded system. Syllabic companding is similar to the action of an automatic volumecontrolinthatthestep sizeis changedslowlyand shouldideallybeproportionedtotheaveragepower of the inputsignal.Syllabiccompandinghasonlybeenappliedto deltamodulationbut, as isshowninthispaper,itcanbe applied t o PCM as well. Paper approved by the Associate Editor for Communication Theory of the IEEE Communications Society for publication after presentation at the 1972 Electronic Instrumentation Conference, Hobart, Australia. Manuscript received July 21, 1972; revised January 25, 1973. ’ The author is with the James Cook University of North Queensland, Townsville, Australia.

memory, corresponding to a control wordof 2 b. Thispaperpresentsanewapproachtothedesignofthe companding strategy which enables one t o select control words, or memory, of any length, thereby allowing one t o select the ratio of the attack and decay time constants of the companding. The digital syllabic companding delta modulation (DSCDM) hardware discussed in this paper has a companding ratiowhich is about 20 dBmorethanhaspreviouslybeen possible. The input power to a code modulator can be normalized by dividingit bythesquareofthestepsize.Theresulting normalizedinputpower is averyusefulparameterforthe is ameasure of overload. design of the companding since it The corresponding term in amplitude modulation is modulationdepth.Foracodemodulatorwithoutcompandingthe step sizeis fixedandthenormalizedinputpower is thus directly proportional to the input power. 11. DIGITALCOMPANDINGPRINCIPLES

Fig. 2 showsablockdiagram of a codemodulatorwith companding.Thecodegeneratorgeneratesdifferentcontrol words until the correct one is obtained, which is then transmitted. A measure of the normalized input power is detected and this is used to control the step size store which in turn controls the multiplier. Provided no transmission errors have occurred, the step size at the transmitter and receiver will be the same. In order to derive a measure of the normalized input power, one must select one or more binary patterns or control words, therelativeoccurrenceofwhich varies with the normalized input power. The best control can be achieved if the function of relative occurrence of the control word versus normalized input power is a monotonic increasing or decreasing function. Fig. 3 shows a typical graph of the relative occurrence of a suitablecontrolwordversusnormalizedinputpower.The

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